The Whole Number, Filozofia, Filozofia - Artykuły
[ Pobierz całość w formacie PDF ]
Mind Association
The Whole Number
Author(s): G. Frege
Source: Mind, New Series, Vol. 79, No. 316 (Oct., 1970), pp. 481-486
Published by: Oxford University Press on behalf of the Mind Association
Stable URL:
Accessed: 26/10/2008 10:44
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
. JSTOR's Terms and Conditions of Use provides, in part, that unless
you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you
may use content in the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the
scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that
promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org.
Oxford University Press
and
Mind Association
are collaborating with JSTOR to digitize, preserve and extend
access to
Mind.
VOL. LXXIX
No.
316]
[October,
1970
M
IND
A QUARTERLY REVIEW
OF
PSYCHOLOGY AND PHILOSOPHY
I.-THE WHOLE NUMBER
BYG.
FREGE
[Translator's
Note: " Le Nombre Entier " appeared in Revue de Meta-
physiqueet de
Morale,
vol. iii
(1895), pp.
73-78. It is
written in French.
The translation is
by
V. H. Dudman
(University
of
New South Wales).]
I
HAVE
noticed that this Journal is trying to reconcile mathe-
matics and philosophy, and to me this seems very valuable.
Indeed, these sciences cannot but profit by exchanging ideas.
It is this that prompts me to enter the discussion. The views
put forward by M. Ballue in the May number' are doubtless
shared by most mathematicians. But they embody logical diffi-
culties which to me seem serious
enough
to be worth
exposing-
all the miorebecause
they might
obscure the
issues, and make
philosophers stop bothering about the principles of arithmetic.
To begin with, it seems worth pointing out that a
frequent
fault
of mathematicians is their
mistaking symbols
for the
objects of
their investigations. In fact, symbols are only the means-
albeit very useful: indispensable, even-of
investigation,
not
its
objects.
These latter are
represented by the symbols.
The shapes
of the
signs,
and
their physical and chemicalproperties,
can be more or less appropriate, but they are not essential.
There is no symbol that cannot be replaced by another of dif-
ferent shape and qualities, the connection between things and
symbols being purely conventional. This goes for any system
of signs and for
any language. Language
is a
powerful
instrument
of the human intelligence, no doubt: but one
language
can be as
useful as another. So it is necessary not to overrate words and
1
Le Nombreentierconsiderecommefondementde
l'analyse mathgmatique.
16
481
482
G.
FREGE:
symbols, either by ascribing to them quasi-magical powers over
things, or by mistaking them for the actual things of which they
are at most the (more or less accurate) representations. It
hardly seems worth insisting on the point, but M. Ballue's article
is perhaps not immune from the error in question. His topic
is whole numbers. What are they? M. Ballue says:
"
Pluralities
are represented by symbols called whole numbers."
According
to him, then, whole numbers are symbols, and it is of these
symbols that he wants to speak. But symbols arenot, and cannot
be the foundation of mathematical analysis. When I write
down 1+ 2 = 3 I am putting forward a proposition about the
numbers 1, 2 and 3, but it is not those symbols that I am talking
about. I could substitute A, B and
r
for them; I could
write p instead of + and e instead of
=.
By writing ApBWrI
should then express the same thought as before-but by means
of different symbols. The theorems of arithmetic are never
about symbols, but about the things they represent. True,
these objects are neither palpable, visible, nor even real, if
what is called real is what can exert or suffer an influence.
Numbers
do not undergo change, for the theorems of arithmetic
embody
eternal truths. We can say, therefore, that these objects
are outside time;
and from this it follows that they are not
subjective percepts
or ideas, because these are continually
changing
in
conformity
with psychological laws. Arithmetical
laws
form no part of psychology. It is not as if every man had
a number
of his
own,
called one, forming part of his mind or his
consciousness:
there is
just
one number of that name, the same
for everybody,
and
objective. Numbers
are therefore very
curious objects, uniting
in themselves the apparently contrary
qualities
of
being objective
and of being unreal. But it emerges
from
a
more
serious
consideration
that there is no contradiction
here. Negative numbers,
fractions, etc. are of the same nature;
and
perhaps
that is why in arithmetic too much store is often
set by
the
symbols.
Because of the difficulty of identifying
objects which are neither discernible to the sense
nor
psycho-
logical, visible objects have been substituted for them.
But
this is to forget that these symbols are not what we want
to
study. And so numbers have a double nature conferred upon
them: they are called symbols; and yet they are themselves
represented-they are given names. M. Ballue writes:
"
Like
all symbols, the whole number admits of a double representation:
the sound which it producesupon the ear, the impressionwhich its
written name produces upon the sight . .. Besides this, the whole
number
possesses
its own particular written representation,
THE WHOLE NUMBER
483
requiring the use of special characters called figures. The aim
of numeration is to study the means of representing all whole
numbers with a small number of words and figures." What is
it then that the figure 2 designates? A number; which is to say,
a symbol, according to M. Ballue. Is it the word deux? If that
were the case, we Germans should have numbers which were
different from those of Frenchmen, and our arithmetic would
be a different science from theirs, having different objects of
investigation. Perhaps M. Ballue's opinion is that the word
deux represents the same number as the figure 2. But whatever
this number might be, it represents a plurality, and is itself
represented by the figure 2. What then do we want with this
somewhat mysterious intermediary? Why will it not do to
designate the plurality directly by the figure?
It might be thought that this is only a verbal slip on M.
Ballue's part, which could easily be corrected by substituting
plurality for whole number in the title of his article. For it is
pluralities
of
which whole numbers are
the symbolic representa-
tives, according to M. Ballue. But this will not save us from all
the difficulties. What is a plurality? M. Ballue replies:
"
The
assemblage of several distinct objects, considered as distinct,
without attention to the nature or shape of these objects, is
called a plurality. It will be seen that a plurality is an assemblage
of units."
This definition is not as clear as the author seems to think.
The sense of the word plurality could be found contained in
the word severaland in the plural form, but M. BaDlueadds some
restrictions, saying " distinct objects, considered as distinct,
without attention to the nature or shape of these objects ".
What he is calling distinct here he has previously called isolated,
saying:
"
An isolated object, considered as isolated, in abstrac-
tion from its nature or shape, is given the name of unit." It
will perhaps be objected that if the objects were absolutely
isolated there would be no assemblage. And besides, it is to be
doubted whether an absolutely isolated object exists, every
material
particle being
related to every other by gravitation.
So
the
precise degree
of isolation
required
would have to be
specified. I shall not labour
this
point, but
I do want to enquire
more closely into what M. Ballue intends by
the words " con-
sidered as distinct, without attention to
the nature or
shape
of
these objects ", and by the words
"
considered
as
isolated,
in
abstraction from its nature or shape ". What
strikes me here
is that the way of considering an object,
and the abstractions
performed in the mind of a subject, seem to be being taken for
484
(". FREGE:
qualities of the object. I ask: after the object
has been con-
sidered as isolated, is it the
same object as before?-or has one
created a new object by considering
it? In the former case,
nothing
essential would be changed. And surely, if I consider
the planet Jupiter as distinct or isolated,
its gravitational ties
to the other heavenly
bodies do not become any the''weaker;
and if I
abstract from its mass and its spheroidal shape, Jupiter
loses neither its mass
nor its shape. So what is the point of
performing this abstraction?
There would be a psychological
difficulty over this as well.
While I am considering an object,
I can be sure that it is being considered.
But in conducting
a proof I have to fix my attention upon other
objects successively,
for I am incapable of considering each of
even a hundred objects
at the same time. It is all the
more difficult in that, without
getting the objects mixed up, I am not to
attend to their nature
or shape.[?Tr.1] I should thus lose the assurance
that these
objects were all in fact units. Certainly they would not be
units
with respect to me: perhaps with respect to other people they
would but I should probably know nothing of this. And even
if I did know, it would be useless from the point of view of my
proof,
for I could infer nothing thence.
Orion is an assemblage of stars. If it is possible in general
to consider objects as distinct, without attention to the nature
or shape of these objects, then it will be possible in this case.
After performing this consideration we shall say, if we take M.
Ballue at his word, that the constellation is a plurality. And
since the name Orion is a symbol for that plurality, we shall
regard
this word as a number. Admittedly he does not actually
say that the stars are considered as distinct, etc. But that is
neither
herenor there: granted that theconstellation is
aplurality,
the
name
of the constellation is a symbol for a plurality.
Let us examine the alternative conception, that the considered
object
is different from the original object. The sun for instance,
as a material, luminous body,
having a shape and occupying
a position, would be different from the
sun considered as distinct,
in abstraction from its nature or shape.
It
might
be said that
the latter is created by the act of considering it and that,
since
an external object cannot be created in this way, it would
have
to be a subjective idea or something of the sort in the mind of the
person performing that consideration and that abstraction.
By thus considering the"sun, everyone would form such an idea
of his own, distinct from anyone else's. Pluralities would then
1
C'est d'autantplus difficilequeje
ne dois
pas
me
prgoccuper
de la nature
ou de la forme des objectssans les confondre.
[ Pobierz całość w formacie PDF ]